Prove that for `n=1, 2, 3...` `[(n+1)/2]+[(n+2)/4]+[(n+4)/8]+[(n+8)/16]+...=n` where `[x]` represents Greatest Integer Function

Prove that (n!) (n + 2) = [n! + (n + 1)!]

Statement-1: the highest power of 3 in .^(50)C_(10) is 4. Statement-2: If p is any prime number, then power of p in n! is equal to [n/p]+[n/p^(2)]+[n/p^(3)] + . . ., where [*] denotes the greatest integer function.

The value of lim(n->oo)((1.5)^n + [(1 + 0.0001)^(10000)]^n)^(1/n) , where [.] denotes the greatest integer function is:

Prove that 1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^2+6n-1))/3

Prove that, 1^2 + 2^2 + …..+ n^2 gt (n^3)/(3) , n in N

Prove that .^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3) if n gt 7 .

The value of lim_(xtoo^+){lim_(ntooo)([1^(2)(sinx)^(x)]+[2^(2)(sinx)^(x)]+……….+[n^(2)(sinx)^(x)])/(n^(3))} is (wehre [.] denotes the greatest integer function)

Prove that sin(n+1)xsin(n+2)x+cos(n+1)xcos(n+2)x=cosx

Prove that 2^n gt n for all positive integers n.

1+2+3+…….+(n+1)= ((n+1) (n+2))/(2) , n in N .